Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics comma complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory kan lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorAlizter
    • CommentTimeNov 6th 2018

    This is a new article.


    In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.